WARM: Place butter, milk, and water in a 2-cup measuring cup. Heat on high for about 1 minute. The milk mixture should be warm to touch, about 115ºF. Continue to heat in 15-second intervals until the butter melts.
COMBINE: In the bowl of an electric mixer, fitted with the dough-hook attachment, add 4 cups of flour, potato flakes, yeast, sugar, and salt. Start the mixture on the stir setting, stream in the warm milk mixture and the egg.
KICK IT UP: When combined, increase the speed of the mixer to medium-low and allow the mixer to run for 4-5 minutes. Once 2 minutes have elapsed, if the dough is sticking to the sides of the bowl, add ¼ cup of flour at a time until the dough begins to pull away from the sides and forms a ball.
REST: Remove the dough from hook, cover the bowl with plastic wrap and allow the dough to rest for 10 minutes. While the dough is resting, position a rack in the center of the oven and preheat the oven to 180ºF. Spray a 9x13 baking dish with cooking spray, set aside.
ROLLS: Turn the dough out onto a lightly floured work surface. Using a bench scraper, divide the dough out into 12 equal pieces. Fold the seams in and roll into a smooth ball. Place in the prepared baking dish. Repeat with all the dough balls. Using a pastry brush, brush water on the tops of the rolls so they don't dry out in the oven.
PROOF: Place the baking dish in the oven, turn the oven OFF and allow the dough to rise for 20 minutes. DO NOT OPEN THE OVEN DOOR.
PREHEAT: Removing baking dish from oven, place on the counter while you preheat your oven to 375ºF.
BAKE: the dinner rolls for 18-22 minutes or until the tops are nice and golden. If the rolls begin to brown early, just place a piece of foil loosely on the surface. Brush rolls with melted butter when done.